Gravitational question ( calculus )

[peterpang1994]

Homework Statement

2 point mass M₁ and M₂ . Their separation is r₀. They are released at rest at t=0. Find the time of collision (T)of them due to gravitational attraction in terms of M₁, M₂ , G and r₀.

Homework Equations

F=ma
∫a(dt)² = s (a is acceleration, s is displacement)
x₀=(M₂/(M₁ + M₂))r₀ ( x is the position of center of mass respect to M₁ )
F=-GM₁M₂/r²

The Attempt at a Solution

I used ∫a(dt)² = s and make it becomes dt=(v/a)ds. As they should collide at the center of mass, the displacement of M₁ should be equals to x. Therefore,
T=∫(v/a)dx
sup x=(M₂/(M₁ + M₂))r , let (M₂/(M₁ + M₂)) = R
x=Rr
dx=Rdr
T=R∫(v/a)dr (from r=r₀ to r=0)
By law of conservation of energy,
(1/2)M₁v² = GM₁M₂(1/r - 1/r₀)
v=√(2GM₂(1/r - 1/r₀))
a=GM₂/r²
∴T=R∫√(2GM₂(1/r - 1/r₀)) / (GM₂/r²) dr
After some calculation, I got
T=R√(2/(GM₂r₀²))∫r²√(1/r - 1/r₀)dr (from r=r₀ to r=0)
I found that integration is very difficult. I wonder whether I get the correct direction or having conceptional and calculating error. I just hope that any help can be given. Thanks a lot.

Homework Statement

Homework Equations

The Attempt at a Solution

 

[peterpang1994]

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